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From: adiabat <rx@awsomnet.org>
Date: Wed, 10 May 2017 10:59:08 -0400
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To: Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
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Subject: Re: [bitcoin-dev] Per-block non-interactive Schnorr signature
	aggregation
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I messed up and only replied to Russel O'Connor; my response is copied below.
And then there's a bit more.

-----
Aha, Wagner's generalized birthday attack, the bane of all clever tricks!
I didn't realize it applied in this case but looks like it in fact does.
 applies to this case.  It would have to be a miner performing the
attack as the s-value would only be aggregated in the coinbase tx, but
that's hardly an impediment.

In fact, sketching it out, it doesn't look like the need to know m1,
m2... m_n is a big problem.  Even if the m's are fixed after being
chosen based on the P1... Pn's, (in bitcoin, m always commits to P so
not sure why it's needed in the hash) there is still freedom to
collide the hashes.  The R values can be anything, so getting h(m1,
R1, P1) + h(m2, R2, P2)... to equal -h(m0, R0, P0) is doable with
Wagner's attack by varying R1, R2... to get different hashes.

I *think* there is a viable defense against this attack, but it does
make the whole aggregation setup less attractive.  The miner who
calculates s-aggregate could also aggregate all the public keys from
all the aggregated signatures in the block (P0, P1...), sort them and
hash the concatenated list of pubkeys.  They could then multiply s by
this combo-pubkey hash (call it h(c)).  Then when nodes verify the
aggregate signature, they need to go through all the pubkeys in the
block, create the same combo-pubkey hash, and multiply s by the
multiplicative inverse of the h(c) they calculate, then verify s.  I
believe this breaks the Wagner generalized birthday attack because
every h(m_i, R_i, P_i)*h(c) included or omitted affects the c part of
h(m0, R0, P0)*h(c).

I'm not sure how badly this impacts the verification speed.  It might
not be too bad for verification as it's amortized over the whole
block.  For the miner doing the aggregation it's a bit slower as they
need to re-sort and hash all the pubkeys every time a new signature is
added.  Might not be too slow.

I'm not super confident that this actually prevents the generalized
birthday attack though.  I missed that attack in the previous post so
I'm 0 for 1 against Wagner so far :)

-----

Andrew: Right, commiting to all the R values would also work; is there
an advantage to using the R's instead of the P's?  At first glance it
seems about the same.

Another possible optimization: instead of sorting, concatenate all the
R's or P's in the order they appear in the block.  Then have the miner
commit to s*h(c)^1, the multiplicative inverse of the hash of all
those values.  Then when nodes are verifying in IBD, they can just
multiply by h(c) and they don't have to compute the inverse.  A bit
more work for the miner and a bit less for the nodes.

-Tadge